Digital adaptive filtering is widely used in the telecommunication field, particularly for designing equalizers and echo cancellors in modems. A well known technique for achieving adaptive filtering is known as the Least Mean Square (L.M.S.) adaptation process that is also referred to as the "Gradient" method which proved its efficiency and simpleness.
The LMS adaptive filtering process is based on the use of coefficients which are computed in accordance with the following formula: ##EQU2## where e(n) is the estimation error.
The step size .alpha. is used for performing an adjustment of the convergence of the filtering process. Prior art document "Analyse des signaux et filtrage numerique adaptatif" by M. Bellanger, Ed. Masson, 1989, teaches that the value of the step size .alpha. should be computed in accordance with the formula: ##EQU3## where .delta..sup.2.sub.x represents the power of the input signal that is being processed, and .alpha..sub.0 represends a constant.
The skilled man is generally aware that he has to choose the value of .alpha..sub.0 in a range .vertline.0, .alpha..sub.s .vertline. where .alpha..sub.s represents the instability threshold of the system. It can be shown that is .alpha..sub.0 is chosen great (but still inferior to .alpha..sub.s), the convergence speed is high but also with a non satisfactory high residual error. On the contrary, should .alpha..sub.0 be chosen lower, the residual error is rendered lower, but at a cost of a lower convergence speed.
FIGS. 1a and 1b are respectively a one-dimension and two-dimensions graphical representations of the gradient method. More particularly, FIG. 1a shows the representation of the means error e.sup.2 (n) as a function of a single filter coefficient C1. FIG. 1b illustrates the two-dimensional parabolic figure of the mean error e.sup.2 (n) as a function of the two coefficents C1 and C2 characterizing a two-coefficient filter. As shown in FIG. 1b, it appears that from an initial setting ##EQU4## there is a convergence process that makes the vector C.sub.(n) reaching the optimum C.sub.opt which coordinates are C1opt and C2opt. In the theorical representation of FIG. 1b, that any value of .alpha..sub.0 being inferior to .alpha..sub.s ensures the convergence process of C.sub.(n) towards the optimal vector C.sub.opt . However, it should be noticed that this theorical representation is quite satisfactory when the signal being processed has the characteristics of white noise, and appears uncorrelated and stationnary.
However, the real situation is somewhat quite different when voice signals are to be processed by the LSM method. On the contrary to white noise, speech signals are correlated and present local stationarity. The problem of designing equalizers for processing voice signal is addressed in prior art document "Second-order Convergence Analysis of Stochastic Adaptive linear filtering" by Odile Macchi et al, IEEE Trans. on Automatic Control, vol. AC-28, n.sup.o 1, January 83; and, from the same author in document "Convergence Analysis of self-Adaptive Equalizers", IEEE trans. on Information Theory, vol. IT-30, n 2, March 84. Basically, the real situations differ from the theorical and ideal model of FIG. 1b in the existence of some local maxima, that are different from the optimum vector C.sub.opt , and which are likely to disturb the gradient convergence mechanism.
Such real situations are illustrated in FIG. 1c which shows that the residual error might strongly depend on the particular setting conditions ##EQU5## as well as the value of .alpha..sub.0 that is chosen. It was mentioned above, that it was possible to reach the optimum C.sub.opt in the theoretical model with a low value of the residual error. However, in real situations , it appears that a low value of .alpha..sub.0 might in certain circonstances not ensure the convergence towards the optimum vector C.sub.opt because of the existence of the disturbing local optima such as shown in FIG. 1c. Should the filtering mechanism converge towards a local minimum instead of the optimum value, this will result in unsatisfactory value of the residual error. To improve the situation, it is necessary to increase the value of .alpha..sub.0 , thus facilitating the process to diverge from the local maxima which are not the optimum point. However, as mentioned above, the increase of .alpha..sub.0 also causes an increase of the residual error.
Additional difficulties are encountered from the increase of the value of .alpha..sub.0 when processing voice or other non stationary signals in equalizers that are used in the telecommunication field. Indeed, in such signals the actual value of the power .delta..sup.2.sub.x is subject to large fluctuations that actually depend on the vocal message being processed. For instance, vocal signals may exhibit strong and sudden increase in the power of the signal. These strong variations might occur more rapidly relatively to the normal processing of the signal. It should be noticed that the echo cancellors that are generally used in the telecommunication field are based on digital filters that process the signals over 32 or 64 milliseconds, that is to say the samples which are used for filtering process generally exceeds 32 or 64 milliseconds. This set of samples are in a range that extends over 32 or 64 milliseconds, is also used for computing an estimation of the power .delta..sup.2.sub.x. Voice signals that are transmitted on telecommunication lines present the characteristics of requiring filtering process that uses samples that extend over 32 or 64 milliseconds, while the characteristics of speech signals might cause sudden variations--less than 10 milliseconds--in the power of the signal being processed. It therefore results in the fact that in equalizers used in the telecommunication field, the estimated value of .delta..sup.2.sub.x might be too low with respect to the actual and instantaneous power of the signal. This inevitably spoils the .alpha. determination process which is based on the previously mentioned formula: ##EQU6##
Indeed, if .delta..sup.2.sub.x is estimated far lower than the actual power of the signal, the value of .alpha. being computed might exceed the value of .alpha..sub.s being mentioned above, thus resulting in a divergence of the coefficients determination.
As a result, it appears that voice signals are not easily processed by traditional equalizers which are used in the telecommunication field. In both cases, the choice of small and large values of .alpha..sub.0 might not ensure the convergence process of the coefficient determination because of the particular characteristics of the speech signals.